How do I change the 3D View rotation mode? To be more like edit mode
Each of your object's origin points are at the center of the scene instead of the center of the object. In the pivot point mode you have selected (Median Point), objects will be transformed about the median pivot point of all the objects that are currently selected (in your case only one is selected).
To fix the pivot point and have rotation occur as you want it to, go to
Object > Transform > Origin to Geometry
Or one of the other options if you prefer it to be at the center of mass or 3d cursor.